Complete code for my tutorial on mouse dragging in Love2D.

main.lua

function love.load()
  rect = {
    x = 100,
    y = 100,
    width = 100,
    height = 100,
    dragging = { active = false, diffX = 0, diffY = 0 }
  }
end

function love.update(dt)
  if rect.dragging.active then
    rect.x = love.mouse.getX() - rect.dragging.diffX
    rect.y = love.mouse.getY() - rect.dragging.diffY
  end
end

function love.draw()
  love.graphics.rectangle("fill", rect.x, rect.y, rect.width, rect.height)
end

function love.mousepressed(x, y, button)
  if button == "l"
  and x > rect.x and x < rect.x + rect.width
  and y > rect.y and y < rect.y + rect.height
  then
    rect.dragging.active = true
    rect.dragging.diffX = x - rect.x
    rect.dragging.diffY = y - rect.y
  end
end

function love.mousereleased(x, y, button)
  if button == "l" then rect.dragging.active = false end
end

You might wanna use the updated version. Available since LÖVE 0.10.0
https://gist.github.com/a-racoon/1ca3b9f467ed491d404035400dfd8953

Why don't you use the love.mousemoved (x, y, dx, dy, istouch) function ?

Note to anyone using this after Love 0.10.0

The current callbacks use numbers, rather than letter strings to identify the mouse buttons.
So lines that say:
button == "l" -- (lower case 'L')
have to be changed to say:
button == 1 -- (button number 1)